298 
MR. G, T. WALKER OX REPULSION AXD ROTATION 
will have no direct resultant mechanical effect on the cylinder, and it will suffice to 
lake the components of induction due to the external system. 
The components of electromagnetic force along S 57 , <77 89, 8z are 
— /3w, aw, 0, 
and hence the mechanical force on the shell, parallel to the initial line from which 9 is 
measured, is (per unit length) 
f 2ir 
a \ 89 . w ( — (3 cos 9 — a sin 9). 
The force in the perpendicular direction (perpendicular to the axis of the shell) is 
a ( 89 . w (a cos 9-/3 sin 9). 
J 0 
The couple about the axis is 
r 2 tt 
cr 89 . aw 
j 0 
tending to increase 9. 
Now, with the kind of field that we have taken, 
F 0 = 0 , 
G 0 = 0, 
and the period being 27r/p, the value of H 0 over the surface may be expanded in the 
series, 
71= 00 
H 0 = E {[M„ cos n9 + N„ sin n9] sin pt -(- [Q„ cos n9 + R„ sin n9] cos pt}. 
n = 0 
We have already shown (vi.) that when 
H 0 = Ae ine sin pt 
at the surface, then at the surface 
in . . a . 
a 0 = — Ac sin pt. 
Hence, with the expansion of H 0 , we get 
n = 00 yi 
a 0 = E - {[ — M„ sin n9 -fi N„ cos n9] sin pt + [ — Q„ sin n9 -f ~R n cos n9] cos pt}. 
n = 0 « 
or J 
The corresponding value of iv has been found to be (v.) 
n _ ^ 
E — . , „ - {[ 27 Tap sin pt + oni cos pt] cos n9 + N„ sin n9] 
n = x 47 Pay* + <r~n* L 
+ [Strap cos pt — <rn sin pi] [Q„ cos n9 + R,, sin n9]} 
